Standard form y= ax² + bx + c
Vertex (could be minimum or maximum) - the bottom point of a parabola if it opens up, the top point if it opens down. Note the vertex point is referred to by the point (h, k). We use the letters h and k to represent the x and y coordinates.
Optimal Value - The y-coordinate of the vertex
Zeros or Roots - the x-intercepts of the parabola, there could be 0, 1 or 2
Translations of y = x²
Vertical Stretch - when 'a' is greater than 1 or less than -1, the graph is vertically stretched when compared with y = x²
Vertical Compression - when a is between -1 and 1, the graph is vertically compressed when compared with y = x²
Reflection on the x-axis when a is negative, the graph is reflected on the x-axis, and the parabola opens down.
Properties of a Quadratic Relations
A parabola is always symmetric. The axis of symmetry is the vertical line x = h,
The x-co-ordinate of the vertex, h, is always the midpoint of the two zeros. (if there are two zeros).
Graphing a Parabola
Label the vertex, zeros and the y-intercept. Try your best to draw a curve close to a U shape. Don't draw a V-shape.
3.1 Quadratic Relations and Second Differences
Line is in the form y= mx + b
Quadratic is y = Ax² + Bx + C
For a line First Differences are the same, for Quadratics second differences are the same.
A polynomial with degree 2 is a quadratic relation
HW p254 #2, 3, 4, 5, 6, 8
3.2 Factored Form of a Quadratic Equation
y = a(x - s)(x - t)
s and t are the zeros.
'a' determines the direction of opening and whether the parabola is vertically stretched or compressed.
For example is y= 4(x-2)(x+5) the zeros will be 2 and -5, and the axis of symmetry will be -1.5.
If we have the zeroes and one other point we can determine the equation of the parabola.
For example, if zeros are 4 and -2, and another another point is (3, 2) then
y = a(x - s)(x - t)
y = a(x - 4)(x + 2)
substutite in known point
2 = a(3-2)(3+2)
2 = 5a
a = 2/5
y = 0.4(x - 4)(x + 2)
HW. p 230 5, 6 a - n
p267 #7 , 9 , 10, 11, 15
3.3 Determining the Equation of a parabola in factored form
HW: p. 280 # 1-5, 8, 9ace, 11, 12.
Study for First Unit 3 test.
common factoring
second differences
standard and factored form.
how 'a' affects opening up or down
sketching a parabola
finding zeros in factored form
finding equation of axis of symmetry
finding vertex, optimal value, minimum or maximum value
using zeros to find equation in factored form (solving for 'a', when you have zeros and another point)
3.4 How to solve problems of quadratic equations in factored form.
Let x be one of the variables.
Find a quadratic equation to find y, which will be a total like revenue or area.
1. Find the zeros, by setting each factor to zero
2. Find the axis of symmetry by calculating the midpoint of the two zeros.
3. The x value of the axis of symmetry will be part of the solution.
4. Plug x value into equation to find y value which will be the other part of the solution.
Example The cost of a ticket to a hockey arena which seats 800 people is $3. At this price, every ticket is sold. A survey indicates that for every dollar increase in price, attendance will fall by 100 people. What ticket price results in the greatest revenue? What is the greatest revenue?
Let x be the number of price increases.
Revenue is price x quantity
y = (3 + x)(800 - 100x) Find zeros
3 + x = 0 and 800-10x = 0
x = -3 or x =8 (representing, selling tickets at $0, or charging $11 dollars and nobody coming)
midpoint is 2.5, best ticket price is 3 + 2.5 or $5.50
Revenue is price x quantity
y = (3 + x)(800 - 100x)
y = (3 + 2.5)(800 - 100(2.5))
y= (5.5)(550)
y = 3025
Therefore the maximum revenue is $3025
Example
Supermarket cashiers try to memorize current sale prices while they work. A study showed that, on average, the percent, P, of prices memorized after t hours is given approximately by the formula P=-40t +120t. What is the greatest percent of prices memorized, and how long does it take to memorize them?
Need factored form to find zeros
P = -40t2 + 120t
P = -40t(t - 3)
set factors to zero to find zeroes
-40t = 0 and t - 3 = 0
t = 0 and t = 3
midpoint will be t = 1.5 which is axis of symmetry
therefore it will take 1.5 hours to memorized the greatest number of prices. Need to find percentage.
P = -40t2 + 120t
P = -40(1.5)2 + 120(1.5)
P = 90
Therefore she will memorize 90% of the prices in 1.5 hours.
Unit 3- Analyzing and Applying Quadratic Models
Vocabulary for Quadratic Unit
Standard form y= ax² + bx + cVertex (could be minimum or maximum) - the bottom point of a parabola if it opens up, the top point if it opens down.
Note the vertex point is referred to by the point (h, k). We use the letters h and k to represent the x and y coordinates.
Optimal Value - The y-coordinate of the vertex
Zeros or Roots - the x-intercepts of the parabola, there could be 0, 1 or 2
Translations of y = x²
Vertical Stretch - when 'a' is greater than 1 or less than -1, the graph is vertically stretched when compared with y = x²Vertical Compression - when a is between -1 and 1, the graph is vertically compressed when compared with y = x²
Reflection on the x-axis when a is negative, the graph is reflected on the x-axis, and the parabola opens down.
Properties of a Quadratic Relations
A parabola is always symmetric. The axis of symmetry is the vertical line x = h,The x-co-ordinate of the vertex, h, is always the midpoint of the two zeros. (if there are two zeros).
Graphing a Parabola
Label the vertex, zeros and the y-intercept. Try your best to draw a curve close to a U shape. Don't draw a V-shape.3.1 Quadratic Relations and Second Differences
Line is in the form y= mx + b
Quadratic is y = Ax² + Bx + C
For a line First Differences are the same, for Quadratics second differences are the same.
For example y = x² - 3x + 2
A polynomial with degree 2 is a quadratic relation
HW p254 #2, 3, 4, 5, 6, 8
3.2 Factored Form of a Quadratic Equation
y = a(x - s)(x - t)
s and t are the zeros.
'a' determines the direction of opening and whether the parabola is vertically stretched or compressed.
For example is y= 4(x-2)(x+5) the zeros will be 2 and -5, and the axis of symmetry will be -1.5.
If we have the zeroes and one other point we can determine the equation of the parabola.
For example, if zeros are 4 and -2, and another another point is (3, 2) then
y = a(x - s)(x - t)
y = a(x - 4)(x + 2)
substutite in known point
2 = a(3-2)(3+2)
2 = 5a
a = 2/5
y = 0.4(x - 4)(x + 2)
HW. p 230 5, 6 a - n
p267 #7 , 9 , 10, 11, 15
3.3 Determining the Equation of a parabola in factored form
HW: p. 280 # 1-5, 8, 9ace, 11, 12.Study for First Unit 3 test.
Common Factoring
Lessons on common factoring can be found on a mathpower site Math Power Tutorialsspecifically http://www.mathpower.com/tut374.htm
3.4 How to solve problems of quadratic equations in factored form.
Let x be one of the variables.Find a quadratic equation to find y, which will be a total like revenue or area.
1. Find the zeros, by setting each factor to zero
2. Find the axis of symmetry by calculating the midpoint of the two zeros.
3. The x value of the axis of symmetry will be part of the solution.
4. Plug x value into equation to find y value which will be the other part of the solution.
Example The cost of a ticket to a hockey arena which seats 800 people is $3. At this price, every ticket is sold. A survey indicates that for every dollar increase in price, attendance will fall by 100 people. What ticket price results in the greatest revenue? What is the greatest revenue?
Let x be the number of price increases.
Revenue is price x quantity
y = (3 + x)(800 - 100x)
Find zeros
3 + x = 0 and 800-10x = 0
x = -3 or x =8 (representing, selling tickets at $0, or charging $11 dollars and nobody coming)
midpoint is 2.5, best ticket price is 3 + 2.5 or $5.50
Revenue is price x quantity
y = (3 + x)(800 - 100x)
y = (3 + 2.5)(800 - 100(2.5))
y= (5.5)(550)
y = 3025
Therefore the maximum revenue is $3025
Example
Supermarket cashiers try to memorize current sale prices while they work. A study showed that, on average, the percent, P, of prices memorized after t hours is given approximately by the formula P=-40t +120t. What is the greatest percent of prices memorized, and how long does it take to memorize them?
Need factored form to find zeros
P = -40t2 + 120t
P = -40t(t - 3)
set factors to zero to find zeroes
-40t = 0 and t - 3 = 0
t = 0 and t = 3
midpoint will be t = 1.5 which is axis of symmetry
therefore it will take 1.5 hours to memorized the greatest number of prices. Need to find percentage.
P = -40t2 + 120t
P = -40(1.5)2 + 120(1.5)
P = 90
Therefore she will memorize 90% of the prices in 1.5 hours.
HW: p. 282 # 6, 14, 16, 19